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Superposition Modulation: Myths and Facts
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26052012, 03:58 PM
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Superposition Modulation: Myths and Facts
[23198] INTRODUCTION Since Shannon’s landmark contribution in 1948, it has been known that any communication channel is characterized by a socalled channel capacity. Channel capacity is, in simple words, the maximum achievable throughput in bits per channel symbol (called transmission rate) subject to the constraint of quasierrorfree transmission. Operating near capacity implies power efficiency and simultaneously bandwidth efficiency. A transmission scheme is called power efficient if the average transmit power per information bit for a given quality constraint is sufficiently small. Discussions on “green radio,” for example, indicate that power efficiency is becoming even more important nowadays. A transmission scheme is called bandwidth efficient if many information bits can be transmitted per time unit in a given bandwidth. Particularly in wireless communications, we experience that bandwidth is sparse and becoming really expensive. THE PRINCIPLE OF SUPERPOSITION MODULATION In order to explain the principle of superposition modulation, let us consider the summation of two random sequences with elements +1 and –1, respectively. For example, the first sequence is [+1, –1, +1, –1], the second sequence [+1, –1, –1, +1]. Each sequence is referred to as a layer. Trivially, the sum is [+2, –2, 0, 0] if both sequences are equally weighted by one. Given a received value of ±2, the mapping can uniquely be reversed. However, if the received value is 0, detection is ambiguous, because it is unclear whether +1 –1 or –1 + 1 has been superimposed. As shown later, SM with realvalued equal weighting is always nonbijective. In our example N = 2 bits are mapped onto just M = 3 signal points, where M < 2N. PROPERTIES OF SUPERPOSITION MODULATION Given these preliminary remarks, we are now ready to formally define SM using complex baseband notation [6, 10]. As shown in Fig. 1, a binary data stream is first split into N parallel binary data streams. Let bn ∈ {0, 1} denote the nth information bit for a certain time slot, 1 ≤ n ≤ N. All bits bn are mapped onto binary antipodal symbols dn ∈ {+1, –1}. Afterward, the symbols dn are weighted by a set of complexvalued factors αn ∈CI in order to obtain N chips cn in parallel. This weighting corresponds to power and phase allocation and is crucial for the performance of SM. EQUAL POWER ALLOCATION Considering power normalization, in the case of equal power allocation (EPA) the weighting factors are fixed according to αn = 1/√ — N for all 1 ≤ n ≤ N. It can easily be proven that the number of distinct symbols x (i.e., the cardinality of the symbol alphabet X) is equal to ⎪X⎪ = N + 1. Since N + 1 < 2N, SMEPA is always nonbijective. An interesting interpretation is that SMEPA incorporates a lossy source encoder, a fact that is explored later. The data symbols are binomial distributed. For N → ∞, the binomial distribution approaches a Gaussian distribution. GROUPED POWER ALLOCATION So far, we observed that SMEPA is nonbijective. Due to nearGaussian quadrature components, no active shaping is necessary for achieving the capacity. The main bottleneck, however, is that H(X) ~ log2(N). In contrast, SMUPA is bijective, but not capacity achieving and hence not further elaborated. 
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